What Is The Correct Step To Solve For The Variable In The Equation $15 = M - 5$?A. Add 15 To Both Sides. B. Subtract 15 From Both Sides.
Understanding the Equation
When solving for a variable in an equation, it's essential to isolate the variable on one side of the equation. In the given equation, $15 = m - 5$, we need to isolate the variable 'm'. To do this, we'll use the properties of equality, which state that if two expressions are equal, we can perform the same operation on both sides of the equation without changing the equality.
Isolating the Variable
To isolate the variable 'm', we need to get rid of the constant term '-5' that is being subtracted from 'm'. We can do this by adding 5 to both sides of the equation. This will cancel out the -5 on the right-hand side, leaving us with just 'm' on the right-hand side.
Step-by-Step Solution
Let's break down the solution step-by-step:
Step 1: Add 5 to both sides of the equation
Add 5 to both sides:
This simplifies to:
Step 2: Check the solution
Now that we have isolated the variable 'm', we can check our solution by plugging it back into the original equation:
Substitute m = 20:
This simplifies to:
Since the equation holds true, our solution is correct.
Conclusion
In conclusion, to solve for the variable 'm' in the equation $15 = m - 5$, we need to add 5 to both sides of the equation. This will cancel out the -5 on the right-hand side, leaving us with just 'm' on the right-hand side. By following this step-by-step solution, we can confidently say that the correct answer is A. Add 15 to both sides is incorrect, the correct answer is to add 5 to both sides.
Frequently Asked Questions
Q: Why can't we subtract 15 from both sides of the equation? A: Subtracting 15 from both sides would change the equation, and we would not be isolating the variable 'm' correctly.
Q: What if the equation was $15 = m + 5$? How would we solve for 'm' in this case? A: In this case, we would subtract 5 from both sides of the equation to isolate the variable 'm'.
Additional Tips and Tricks
- When solving for a variable, always follow the order of operations (PEMDAS) to ensure that you are performing the correct operations.
- Use the properties of equality to your advantage by performing the same operation on both sides of the equation.
- Check your solution by plugging it back into the original equation to ensure that it holds true.
Real-World Applications
Solving for variables is a fundamental concept in mathematics that has numerous real-world applications. For example, in physics, we use equations to describe the motion of objects, and solving for variables is essential to understanding the behavior of these objects. In economics, we use equations to model the behavior of markets, and solving for variables is crucial to making informed decisions.
Conclusion
In conclusion, solving for variables is a critical concept in mathematics that has numerous real-world applications. By following the step-by-step solution outlined in this article, we can confidently say that the correct answer is to add 5 to both sides of the equation. Remember to always follow the order of operations, use the properties of equality, and check your solution to ensure that it holds true.
Q&A: Solving for Variables
Q: What is the first step in solving for a variable in an equation?
A: The first step in solving for a variable is to understand the equation and identify the variable that needs to be isolated. Once you have identified the variable, you can begin to solve for it by using the properties of equality.
Q: What are the properties of equality?
A: The properties of equality state that if two expressions are equal, you can perform the same operation on both sides of the equation without changing the equality. This means that you can add, subtract, multiply, or divide both sides of the equation by the same value without changing the equality.
Q: How do I isolate a variable in an equation?
A: To isolate a variable in an equation, you need to get rid of any constants or other variables that are being added or subtracted from the variable. You can do this by performing the opposite operation on both sides of the equation. For example, if the equation is $x + 3 = 5$, you can subtract 3 from both sides to isolate the variable x.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation $x + 3 = 5$ is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.
Q: How do I solve a quadratic equation?
A: To solve a quadratic equation, you can use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. You can also use factoring or the quadratic formula to solve a quadratic equation.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that is used to solve quadratic equations. It is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic equation.
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. You can then simplify the expression and solve for x.
Q: What is the difference between a system of equations and a single equation?
A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation, on the other hand, is a single equation that is solved independently.
Q: How do I solve a system of equations?
A: To solve a system of equations, you can use substitution or elimination to solve for the variables. You can also use graphing or matrices to solve a system of equations.
Q: What is the difference between a linear system and a nonlinear system?
A: A linear system is a system of equations in which the highest power of the variable is 1. A nonlinear system, on the other hand, is a system of equations in which the highest power of the variable is greater than 1.
Q: How do I solve a nonlinear system?
A: To solve a nonlinear system, you can use numerical methods or approximation techniques. You can also use graphing or matrices to solve a nonlinear system.
Conclusion
In conclusion, solving for variables is a critical concept in mathematics that has numerous real-world applications. By understanding the properties of equality, isolating variables, and using the quadratic formula, you can solve a wide range of equations and systems. Remember to always follow the order of operations, use the properties of equality, and check your solution to ensure that it holds true.
Additional Resources
- For more information on solving for variables, check out the following resources:
- Khan Academy: Solving Equations
- Mathway: Solving Equations
- Wolfram Alpha: Solving Equations
Frequently Asked Questions
Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2.
Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. You can also use factoring or the quadratic formula to solve a quadratic equation.
Q: What is the quadratic formula? A: The quadratic formula is a formula that is used to solve quadratic equations. It is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic equation.
Q: How do I use the quadratic formula? A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. You can then simplify the expression and solve for x.
Q: What is the difference between a system of equations and a single equation? A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation, on the other hand, is a single equation that is solved independently.
Q: How do I solve a system of equations? A: To solve a system of equations, you can use substitution or elimination to solve for the variables. You can also use graphing or matrices to solve a system of equations.
Q: What is the difference between a linear system and a nonlinear system? A: A linear system is a system of equations in which the highest power of the variable is 1. A nonlinear system, on the other hand, is a system of equations in which the highest power of the variable is greater than 1.
Q: How do I solve a nonlinear system? A: To solve a nonlinear system, you can use numerical methods or approximation techniques. You can also use graphing or matrices to solve a nonlinear system.